Prime numbers are numbers that are only divisible by themselves and one. Ancient Greek mathematicians first studied them. Euclid proved that there are infinite prime numbers. In the "Fundamental Theorem of Arithmetic", Euclid prooved that every integer can be written as a product of primes.

During the beginning of the 17^th century, Fermat proved that every prime number of the form 4n + 1 could be written as the sum of two squares. Fermat also stated that the numbers 2ⁿ + 1 are always prime if n is a power of 2. Numbers that have this property were called Fermat numbers. After 100 years, Euler prooved that this formula does not always work because 2³² + 1 is equal to 4,294,967,297, which is not prime (it is divisible by 641).

Marin Mersenne studied numbers of the form 2ⁿ – 1. They were named after him. Mersenne numbers were not always prime. So far, 37 Mersenne numbers have been found. The largest prime number is 2^3,021,377 - 1, which was found by Ronald H. Clarkson, a sophomore at California State University Dominguez Hills. The 37^th Mersenne prime has 909,526 digits.

Some questions are still unanswered that relate to prime numbers:

1. The Twin Primes Conjecture that there are infinitely many pairs of primes only 2 apart.
2. Goldbach’s Conjecture that every integer greater than two can be written as the sum of two primes.
3. Are there infinitely many primes of the form n² + 1?
4. Is there always a prime between n² and (n + 1) ²?
5. Are there infinitely many prime Fermat numbers?
6. Are there infinitely long progressions of consecutive primes?
7. Are there infinitely many sets of three consecutive primes in arithmetic progression?
8. n² – n + 41 is prime for 0 £ n £ 40. Are there infinitely many primes of this form?
9. Are there infinitely many primes of the form n# + 1?
10. Are there infinitely many primes of the form n# - 1?
11. Are there infinitely many primes of the form n! + 1?
12. Are there infinitely many primes of the form n! – 1?
13. If p is prime, is 2^P – 1 always square free?
14. Does the Fibonacci sequence contain an infinite number of primes?

1.) What is the sum of the first 5 prime numbers?

28
18
25

2.) Which of the following is a prime number?

57
49
61

3.) How many primes are there that are less than 20?

8
12
2

4.) Which is not a prime number?

21
37
19

5.) A prime number is:

a number that is only divisible by itself and one
a number that is even and is divisible by 12 and 18
a number that is odd

6.) Who was (or were) the first to study prime numbers?

Euclid
Fermat
ancient Greek mathematicians

7.) Fermat numbers are:

prime numbers that are even
prime numbers that were discovered by Fermat
prime numbers that have the property of 2ⁿ + 1 when n is a power of 2